Equation solver



April 14, 1964 D. c. THOMAS ETAL 3,129,325

EQUATION SOLVER Filed May 25, 1959 3 Sheets-Sheet 1 c I1 0. mpg ER COM UT X=F(I1..IN) o X IN FEEDBACK EQUATION SOLVER IN=G(X) INPUT COMPUTER T I X=HI,1=/IIdt+D I 1 0 1 2 O C INTEGRATION CONSTANT D 1 1 I I LOG 10g I1 1 GENERATOR I ANTI LOG GENERATOR I I LOG 2 2 GENERATOR c C C4 C5 FEEDBACK E UATION SOLVER fi' X I2=K(X) IN VEN TORS DONALD (J. THOMAS Ammvn April 14, 1964 D. c. THOMAS ETAL 3,129,325

EQUATION SOLVER Filed May 25, 1959 3 Sheets-Sheet 3 IN 50 I ZP KILO F EET T IN DEGREES F l l l I N v INVENTORS DONALD C. THOMAS"-v THOMAS C. HORTH ATTORNEY United States Patent 3,129,325 EQUATION SOLVER Donald C. Thomas and Thomas C. Horth, Los Angeies County, Calif., assignors to Tasker Instruments Corporation, Hollywood, Calif.

Filed May 25, 1959, Ser. No. 815,735 Claims. (Cl. 235-180) This invention relates to an equation solver, and more particularly to an equation solver for solving an equation wherein one or more variables of the equation are related to the solution thereof by independent functions.

In prior art equation solvers it has been the practice to generate input signals representing the variables in an equation by external generator means, and to apply these signals to the input terminals of the equation solver. The input signals were combined in the equation solver in accordance with the equation to produce an output signal representing the solution thereof. Examples of these prior art equation solvers can be found on pages 37 and 39 of Electronic Analog Computers by Korn and Korn, Second Edition, published by McGraw-Hill in 1956. In the charts on pages 37 and 39 of the above-noted publication it will be seen that each equation solver has separate input terminals for every variable in the equation to be solved. It will be understood that signals representing each variable must be generated by external generator means and applied to the input terminals in order to produce a signal representing the solution to the equation at the output terminal of the equation solver.

In accordance with the present invention it has been found that a considerable simplification can be achieved in the input signal generator means by deriving some of the equation solver input signals from the output signal thereof in cases where some of the input signals are related to the output signal by independent functions. In these cases it has been found that the related input signals can be derived quite simply from the output signal by a feedback equation solver running from the output terminal of the equation solver to the corresponding input terminal thereof. The external input generator means for the related input signals are thus entirely eliminated, and replaced by a feedback equation solver which is simpler and more reliable in operation than the input generator means previously employed.

The invention may find application wherever the solution to an equation requires the introduction of an input variable which is a function of the solution. Thus the invention is applicable whenever one or more of the in put variables is related to the solution through independent functions. It will be understood, of course, that at least one of the input variables must be obtained from an external source. In the description which follows an exemplary case will be considered where the equation is definitive of pressure altitude Z which is intended to signify a measurement of altitude in terms of pressure and temperature. Thus the term pressure altitude will be used hereinafter to signify a specific type of altitude determination computed from an equation which may be in the form of (1) below. It will be understood, of course, that the generic concept of the invention is not limited to this particular equation. The important thing to note in this function, however, is that two of the variables are related to Z by independent functions,

T-Tp T 3,129,325 Patented Apr. 14, 1964 ice Z by feedback equation solvers embodying the functions disclosed in FIGS. 3A and 3B.

The method of this invention, however, is not limited to Equation 1 above, but is equally applicable to any equation wherein some of the variables are related to the solution by independent functions. In mathematical terms, this invention is applicable to any equation having the form S=F(I I where at least one variable I can be expressed as an independent function of the solution S, i.e. I =G(S) where X is smaller than N and 6(5) is independent of F(I I Accordingly, one object of this invention is to provide a novel equation solver wherein one or more input signals are derived from an output signal by means of feedback equation solvers.

Another object of this invention is to provide an equation solver having simpler input signal generator means than heretofore known in the art.

An additional object of this invention is to provide a feedback equation solver circuit with which any desired function of an output signal may be derived to produce an input signal related thereto.

A specific object of this invention is to provide a novel equation solver for determining pressure altitude from input signals representing observed temperature, true altitude, and observed relative humidity.

Another specific object of this invention is to provide a novel equation solver for determining pressure altitude wherein input signals representing pressure and standard temperature are derived from an output signal representing pressure altitude by feedback equation solvers.

A further object of this invention is to provide an equation solver simpler, more reliable, and less expensive than any heretofore known in the art.

Other objects and advantages of the invention will be apparent to those skilled in the art from the following detailed description of several illustrative embodiments thereof, in connection with the attached drawings, in which:

FIG. 1 is a block diagram of a general embodiment of the invention.

FIG. 2 is a block diagram of a first specific embodiment of the invention.

FIG. 3 is a block diagram of a second specific embodiment of the invention.

FIG. 3A is a curve showing the relation of the output signal of the embodiment in FIG. 3 to one input signal thereof.

FIG. 3B is a graph showing the relation of the output signal of the embodiment in FIG. 3 to another input signal thereof.

FIG. 3C is a schematic diagram of a feedback equation solver as employed in the invention.

FIG. 1 shows a general embodiment of the invention for solving the equation X=F(I I where I =G(X) Where the function G(X) being independent of the function F(I I Input signals represent-. ing the variables I through 1 are applied to a computer C which combines the input signals according to the function F(I I to produce an output signal which would represent X if a signal representing I were applied to computer C. An input signal representing I is derived from the output signal of the computer C by a feedback equation solver E, which derives the value for I according to the function G(X).

FIG. 2 shows a first specific embodiment of the invention for solving the equation where I =K(X) with K(X) being independent of H(I 1 In this embodiment multiplication of the variable is performed by taking the logarithms thereof in log generators I and I adding the logarithms in a summing amplifier C and taking the antilog in an antilog generator C The integration is performed in an integrator C As in the embodiment of FIG. 1, an input signal representing I is derived from the output signal X by a feedback equation solver E in accordance with the function K(X) FIG. 3C shows a feedback equation solver circuit which can be adapted to solve any equation between two variables, such as indicated by the generalized feedback equations I =G(X) and I =K(X) in FIGS. 1 and 2. The feedback equation solver comprises an operational amplifier A with an input terminal I, an input impedance R and a function generating output circuit terminating in an output terminal 0. The function generating output circuit contains a plurality of individual stages F through F coupled in parallel to the output of amplifier A through a bias network R through R The outputs of the individual stages F through F are summed in a summing circuit S to give an output signal 0. Each individual stage contains a pair of diodes D and D coupled through a common load resistor R to a bias voltage B. One diode is coupled to the output of amplifier A, and the other to a feedback impedance R which is shunted by a feedback capacitance C The feedback impedances are returned to the input of amplifier A.

Each of the individual stages F through F is operative during a predetermined increment of the output voltage from amplifier A to produce a segment of an output curve, the segments being added in summing circuit S to give a complete output curve. By appropriate selection of circuit values any desired function can be approximated with this function generating circuit, as more fully explained in copending application Serial Number 807,- 608 filed April 20, 1959 for Function Generator Circuits by Wendell B. Sander now Patent No. 2,976,430. As used in connection with the embodiments of this invention shown in FIGS. 1 and 2 the input terminal I of the feedback equation solver is coupled to output terminals X, with output terminal 0 of the feedback equation solver coupled to input terminals I and 1 respectively. The circuit values for the feedback equation solver are chosen to solve the equations I =G(X) and I =K(X), respectively. The feedback equation solver circuit can also be used in the embodiment of FIG. 2 for log gen erators C and C and for antilog generator C FIG. 3 shows a second specific embodiment of the invention for solving the Equation 1:

Z TP ELF] o 1 zp z Dom: T +037 Zdt where:

Z =Pressure altitude (where altitude is measured as a function of pressure and temperature) Z-=True altitude D =Altitude correction at reference altitude T=Observed temperature T =Standard temperature E =Saturation vapor pressure F=Observed relative humidity P=Pressure Equation 1 can be rewritten in the form:

o o E o] 2 z,. z-.D0 [z z +0.37 P [a which is more convenient to mechanize than Equation 1.

In Equations 1 and 2 two variables, P and T9, are related to Z by independent functions as shown in FIGS 3A and 3B, which were derived from standard atmosphere data published by the US. Weather Bureau. Therefore, in accordance with the method of this invention these variables need not be generated externally, but can be derived from the signal representing Zp.

Referring now to FIG. 3, Equation 2 is mechanized in a computer having external signal inputs representing T, Z, and F; with input signals representing Log T and Log P derived from an output signal representing Z by feedback equation solvers E and E The variable Z is derived from Z by a differentiating circuit C which may be an operational amplifier with a capacitance input and resistance feedback. The variables are combined in accordance with Equation 2 by taking the logarithms thereof in log generators C through C adding the logarithms in summing amplifiers C and C11, and taking the anti-logarithms thereof in antilog generators C and C Integration is performed in integrator C which may be any suitable integrating circuit, and the final sum is taken in summing amplifier C The feedback equation solver circuit disclosed in FIG. 3C may be utilized for feedback equation solvers E and E and also for log generators C through C and for antilog generators C and C The summing amplifiers may be any suitable summing circuit.

From the foregoing description it will be apparent that the present invention provides a novel equation solver wherein the input signal generator means therefore is simplified by utilizing feedback equation solvers to derive some of the input signals from the output signal produced thereby. It will also be apparent that this invention provides a feedback equation solver which can be adapted to derive an input signal from an output signal in accordance with any relation therebetween; and, therefore, provides a simple and reliable equation solver for determining pressure altitude from input signals representing observed temperature, true altitude, and observed relative humidity.

It should be understood that this invention is not limited to the particular structures and uses disclosed in this application, and that many modifications may be made without departing from the basic teaching of this invention. The method of deriving input signals to equation solvers from output signals thereof has wide application in equation solvers generally, and this invention includes all equation solvers falling within the scope of the following claims.

We claim:

1. In a system of the character described, the combination comprising: an input computer responsive to input signals representing variables 1 I for producing an output signal in accordance with the function F(I I said computer including at least two log generators, means for adding the output signals of said two log generators, and means for generating the antilog function of the output signal of the adding means, thereby to generate a product function of at least two of said input signals; and a feedback equation solver responsive to said output signal for producing at least one of said input signals, but no more than N -1 thereof, said feedback equation solver including a plurality of networks for producing a corresponding plurality of signal segments to represent a predetermined part of the functional relationship between said output signal and each of the input signals which is to be developed thereby.

2. In a system of the character described, wherein a plurality of input signals representing respective variables are translated into an output signal representing a predetermined function of said variables, at least one of said variables, but less than all thereof, being related to the solution thereof by an independent empirical function, the improvement comprising: a feedback equation solver for receiving said output signal and for producing each of said input signals defined by an independent empirical function, said feedback equation solver including means for developing at least one logarithmic function signal, and said input signals including at least one logarithmetic function signal, permitting multiplication and division through addition and subtraction, respectively.

3. The improvement defined in claim 2, wherein said logarithmic function is the logarithm of a variable representing pressure and is combined negatively with at least one other input signal to divide the pressure variable into the variable represented by the other input signal.

4. The improvement defined in claim 2, wherein said logarithmic function is the logarithm of a variable representing temperature and is combined positively with at least one other input signal to multiply temperature by the variable represented by the other input signal.

5. The improvement defined in claim 2, wherein two logarithmic functions representing temperature and pressure are present and are combined positively and negatively, respectively, with said input signals to produce said output signal.

6. An equation solver for solving an equation having the form X =F (I I wherein one variable I is related to X according to the equation I =G(X) and wherein said function F (1 I includes a term containing the product of said one variable I and a second variable I said equation solver including means for generating a signal representing Log I and the equation solver including summing means for adding said signal representing Log I to a signal representing Log 1 to form a signal representing the sum Log l -l-Log I the improvement comprising a feedback equation solver adapted to derive a signal representing Log I from the equation solver output signal according to the equation Log I =L0g[G(X)], and said signal representing Log I coupled to said summing means to form a signal representing Log I +Log 1 within said equation solver.

7. An equation solver for generating an output signal representing pressure altitude Z from input signals representing observed temperature T, true altitude Z, and observed relative humidity F: said equation solver comprising an electrical circuit embodying the relation between pressure altitude Z and observed temperature T, true altitude Z, observed relative humidity F, pressure P, and standard temperature T generator means responsive to signals corresponding to F, P, T, T and Z, for producing an output signal representing Z and feedback equation solver means including first and second function generators responsive to signal Z for producing signals corresponding to T and P, each of said function generators including a plurality of line-segment signal generators for approximating the desired relationship between Z and T in the first function generator, and the desired relationship between Z and P in the second function generator.

8. An equation solver as defined in claim 7, and also including means for deriving a signal representing saturation vapor pressure from said input signal representing observed temperature, and means for deriving a signal representing the derivative of altitude with respect to time from said input signal representing altitude.

9. An equation solver as defined in claim 8 wherein said input signal representing pressure is equal to the logarithm of pressure and wherein said signal representing standard temperature is equal to the logarithm of standard temperature; and wherein said electrical circuit includes log generator means operable to produce signals representing the logarithms of observed temperature, true altitude, observed relative humidity, saturation vapor pressure, and the derivative of altitude with respect to time; and wherein said electrical circuit includes a first summing circuit for adding said signal representing the logarithm of standard temperature to said signals representing the logarithms of observed temperature and the derivative of altitude with respect to time; and wherein said electrical circuit includes a second summing circuit for adding said signal representing the logarithm of pressure to said signals representing the logarithms of saturation vapor pressure, observed relative humidity, and the derivative of altitude with respect to time.

10. An equation solver as defined in claim 9 and also including first and second antilog generators coupled to respective summing circuits; and an integrator coupled to said antilog generators and to said signal representing the derivative of altitude with respect to time; and a third summing circuit coupled to said integrator and to said signal representing true altitude; and said third summing circuit being operable to produce an output signal representing pressure altitude.

References Cited in the file of this patent UNITED STATES PATENTS Hagen et a1. Sept. 2, 1958 Blocker Jan. 31, 1961 OTHER REFERENCES 

1. IN A SYSTEM OF THE CHARACTER DESCRIBED, THE COMBINATION COMPRISING: AN INPUT COMPUTER RESPONSIVE TO INPUT SIGNALS REPRESENTING VARIABLES I1 . . . IN FOR PRODUCING AN OUTPUT SIGNAL IN ACCORDANCE WITH THE FUNCTION F(I1 . . . IN), SAID COMPUTER INCLUDING AT LEAST TWO LOG GENERATORS, MEANS FOR ADDING THE OUTPUT SIGNALS OF SAID TWO LOG GENERATORS, AND MEANS FOR GENERATING THE ANTILOG FUNCTION OF THE OUTPUT SIGNAL OF THE ADDING MEANS, THEREBY TO GENERATE A PRODUCT FUNCTION OF AT LEAST TWO OF SAID INPUT SIGNALS; AND A FEEDBACK EQUATION SOLVER RESPONSIVE TO SAID OUTPUT SIGNAL FOR PRODUCING AT LEAST ONE OF SAID INPUT SIGNALS, BUT NO MORE THAN N-1 THEREOF, SAID FEEDBACK EQUATION SOLVER INCLUDING A PLURALITY OF NETWORKS FOR PRODUCING A CORRESPONDING PLURALITY OF SIGNAL SEGMENTS TO REPRESENT A PREDETERMINED PART OF THE FUNCTIONAL RELATIONSHIP BETWEEN SAID OUTPUT SIGNAL AND EACH OF THE INPUT SIGNALS WHICH IS TO BE DEVELOPED THEREBY. 